The study of Schrödinger equations received a great deal of attention from researchers in the past decades because of its vast applications in several areas of mathematics and mathematical physics. In particular, Schrödinger equations arise in quantum field theory and in the Hartree-Fock theory (see [1]}, [2]}, [3]}, [4]}).
Recently, there is a surge of interest to investigate integrodifferential operators of order close to zero and associated linear and nonlinear integrodifferential equations (see [5]}, [6]}, [7]}, [8]}, [9]}, [10]}). In particular, the logarithmic Laplacian and the logarithmic Schr\(\ddot{\text{o}}\) dinger operator are two interesting examples of such class of operators. The logarithmic Schr\(\ddot{\text{o}}\) dinger operator \((\mathcal {I}-\Delta )^{\log }\)  (see [11]}) and the logarithmic Laplacian \(L_{\Delta }\)  (see [5]}) have the similar behavior locally concerning to the singularity of kernels but
the logarithmic Schr\(\ddot{\text{o}}\) dinger operator eliminates the
integrability problem of the logarithmic Laplacian at infinity. To define the logarithmic Schr\(\ddot{\text{o}}\) dinger operator, let us begin with the following observation
\(\lim _{s\rightarrow 0^{+}}(\mathcal {I}-\Delta )^{s}u(x)=u(x),\ \text{for}\ u\in C^{2}(\mathbb {R}^{N} ),\) 
